Two main differences. One, the red dot (typically) does not magnify, second, instead of having to find crosshairs, you are given a red (or green) dot (which might be a horse shoe or a chevron or the like).
The lenses in a telescopic sight have to focus the image and have some fairly rigid geometrical rules on where that geometry is located. Your eye needs to be in the range of that geometry. That area is referred to as the "eye box" and is the area fore and aft and left and right of the scope where you get a clean clear image of the target. If you are too far left/right or front/back (or combinations of those) the scope image will be distorted or obscured.
Mount up behind the same scope a few thousand times and it will be reflex as to where to put your cheek to get your eye in just the right spot. (Which is very much assisted by the height of the rings mounting the scope and where on the action the scope is mounted, too.) As you have seen on your buddy's rifle, getting the right spot makes it pretty effortless. This is not always the case--I know some folk who have scopes set up that only they can use.
Now, the crosshairs, these are referred to, technically, as the reticule. They can be in all sorts of configurations, fine hair; thick-thin; single post; two post; three post Mil-dots, and on and on. What is important is that they are something your eye has to focus upon to superimpose the target upon (this can get way extra technical in which focal plane the reticule impinges upon--that's a different topic). The cross hairs gain you something over "iron" sights--there's only the one thing to focus upon other than the target, and not three (front sight, rear sight, target).
So, because of the magnification, what you can see around the target is limited. Term of art is "Field of view." If it's a paper target, only being able to see
xx feet eight side of a target is not too huge a deal. But it can be a thing if you are trying to scan something less discrete. Like, that tree three down from the fence post.
A red dot skips a bunch of that. You are looking through a tube. Your eye will notice the tube and your brain works to center that on your vision without conscious effort (we humans are cool that way). A dot is presented in the center of that tube, and it hovers there without regard to where you are behind the optic (and you will naturally try and center it). It has no real "focus" (other than any astigmatism in your vision), so you don't have to pick it out in the field of view. That makes it fast.
You get a red dot that you put on the target and that's where the rounds go. It's really reflexive. For "fieldcraft" it has a distinct advantage, you can shoot both eyes open. There's no lag as your shooting eye changes focus.
It can get off the rails complicated, sadly. There are all sorts of arguments about how big the dot ought to be. (It's not a lot of help if the dot is bigger than the target, for instance.) Some of the dots use a chevron, not a dot, and have ranging information stadia in them, too. This can clutter the view, and the stadia are very much limited to ammo type and other features. Also, for those with astigmatism (an irregularity of cornea that can blur focus, sometimes imperceptibly) that round dot can be a star or a sunburst or sawblade shape. This is not insurmountable, but it can take practice.
Now, this is long and detailed (and only an inch deep at that). For 2¢ get the same scope and rings as your buddy has, and start there on the 10/22. Getting the scope adjusted to point where the rounds go is a good way to use ammo, too. Decent link:
https://www.opticsplanet.com/howto/how-to-choose-and-use-a-boresight.html
If you want to spend a dime or two:
https://www.brownells.com/gunsmith-...hter-kits/magnetic-boresighter-prod79734.aspx
Now that 9mm carbine will likely just love a red dot.
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